When an object is thrown into the air, when the object is at its peak height, is the acceleration 0 or is it still free fall, 9.81 m/s^2?

The acceleration is caused by the object's weight. Since it always, has weight, the acceleration toward the earth is always 9.81 m/s^2.

V = 0
a = g = 9.806 m/sec^2 directed downward

Spock's grampa was wrong. The instant accel is still 9.81 m/s^2 downward.

by definition, at peak, the instantaneous change in height is zero. gravity still works, however, and that force will soon cause it to return. the instant accel is zero, but will soon increase